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Grade 7 Proportional Relationships Practice Guide

Learn to compute fractional unit rates, recognize proportional relationships, connect tables, graphs, and equations, and solve multi-step percent problems.

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This guide covers the Grade 7 Ratios and Proportional Relationships expectations, including fractional unit rates, y = kx, graph interpretation, and multi-step percent applications. View the official mathematics standards.

Skill map

Five connected proportional-reasoning skills

Students should be able to identify the same constant relationship in numerical, graphical, algebraic, and real-world forms.

1÷

Fractional unit rates

Divide one fractional quantity by another and interpret the result.

T

Tables

Test whether y divided by x is constant for every nonzero x-value.

G

Graphs

Recognize a straight line through the origin and find its slope as the unit rate.

k

Equations

Write and use y = kx, where k is the constant of proportionality.

%

Percent applications

Solve tax, tips, discounts, markups, commission, interest, and percent error.

Quick navigation

What this guide covers

Concept 1

Compute unit rates involving fractions

A unit rate compares a quantity to one unit. Grade 7 extends this idea to ratios containing fractions or mixed numbers.

Use division, then interpret

  1. Write the rate as quantity A divided by quantity B.
  2. Convert mixed numbers to improper fractions when needed.
  3. Divide by multiplying by the reciprocal.
  4. Simplify and state the unit in words.
  5. Check whether the size of the answer is reasonable.
Concept 2

Test whether a relationship is proportional

A proportional relationship has one constant multiplicative factor connecting every x-value to its y-value.

Proportional table

xyy ÷ x
273.5
4143.5
6213.5

Each ratio y ÷ x equals 3.5, so y = 3.5x.

Not proportional

xyy ÷ x
252.5
4102.5
6162.667

The first two ratios equal 2.5, but 16 ÷ 6 does not.

Three equivalent tests: constant y ÷ x ratio, an equation y = kx, or a straight graph through the origin.
Concept 3

Connect graphs, equations, and meaning

The constant of proportionality k appears as the unit rate in a table, the slope of the graph, and the multiplier in y = kx.

Proportional graph

The line passes through (0, 0). The point (1, 3) shows the unit rate k = 3.

Equation and point meaning

y = 3x
  • k = 3 is the constant of proportionality.
  • (1, 3) means 3 units of y for 1 unit of x.
  • (4, 12) means 4 units of x correspond to 12 units of y.
  • (0, 0) shows that zero x produces zero y.
Concept 4

Use proportional reasoning for percent applications

Percent means per hundred. The same proportional structure supports increases, decreases, fees, interest, and error calculations.

Percent amount

amount = percent × whole

Use the decimal form of the percent.

Increase or decrease

new = original × (1 ± rate)

Use plus for increase and minus for decrease.

Simple interest

I = Prt

P is principal, r is annual rate, and t is time in years.

Commission

commission = rate × sales

Add a base payment only when the problem states one.

Percent error

|estimate − actual| ÷ actual × 100%

The actual value is the denominator.

Multi-step percent

apply each percent to its current base

A discount and later tax are not usually combined by simple subtraction.

Step by step

Eight worked examples

State the constant, equation, or percent structure before carrying out the arithmetic.

1. Fractional unit rate

A cyclist travels 7.5 miles in 1.25 hours.

  1. Rate = 7.5 ÷ 1.25.
  2. Rate = 6.
Answer: 6 miles per hour

2. Unit rate with fractions

A recipe uses 2 1/2 cups for 3/4 of a batch. Find cups per batch.

  1. Convert 2 1/2 to 5/2.
  2. (5/2) ÷ (3/4) = (5/2)(4/3) = 10/3.
Answer: 3 1/3 cups per batch

3. Test a table

The pairs are (3, 8.1), (5, 13.5), and (8, 21.6).

  1. 8.1 ÷ 3 = 2.7.
  2. 13.5 ÷ 5 = 2.7.
  3. 21.6 ÷ 8 = 2.7.
Answer: Proportional; y = 2.7x

4. Find k from a graph point

A proportional graph contains (4, 14).

  1. Use k = y ÷ x.
  2. k = 14 ÷ 4 = 3.5.
Answer: k = 3.5 and y = 3.5x

5. Interpret a point

The equation c = 8t models ticket cost. Interpret (6, 48).

  1. t represents tickets and c represents cost.
  2. Six tickets cost 48 dollars.
Answer: 6 tickets cost $48

6. Discount followed by tax

A $90 item is discounted 20%, then taxed 6%.

  1. Discounted price = 90 × 0.80 = 72.
  2. Final price = 72 × 1.06 = 76.32.
Answer: $76.32

7. Commission with base pay

A salesperson earns $300 plus 7% of $6,400 in sales.

  1. Commission = 0.07 × 6,400 = 448.
  2. Total = 300 + 448.
Answer: $748

8. Percent error

A measured value is 102 cm and the actual value is 100 cm.

  1. Difference = |102 − 100| = 2.
  2. Percent error = 2 ÷ 100 × 100%.
Answer: 2%
Error prevention

Common proportional-reasoning mistakes

Using addition instead of multiplication

A proportional relationship is controlled by a constant multiplier, not a constant added difference.

Checking only one pair

One ratio cannot prove a whole table is proportional. Check every nonzero pair.

Ignoring the origin

A straight graph that does not pass through (0, 0) is not proportional.

Reversing the unit rate

The order of division must match the requested units.

Treating k as an unexplained number

State what k means in the problem context and include units.

Combining percent changes incorrectly

Apply each percent to the correct current amount rather than simply adding or subtracting rates.

Using the estimate in the percent-error denominator

Percent error is measured relative to the actual value.

Dropping money precision

Round currency to the nearest cent at the appropriate final step.

Independent practice

20 Grade 7 proportional relationships questions

Show the constant, equation, or percent structure where appropriate.

A. Fractional unit rates

  1. A runner travels 9 miles in 3/4 hour. Find the unit rate in miles per hour.
  2. A recipe uses 2 1/2 cups of flour for 3/4 of a batch. Find cups of flour per full batch.
  3. A 5/6-pound package costs $7.50. Find the price per pound.
  4. A painter uses 3/8 gallon of paint for 3/4 of a wall. Find gallons per wall.

B. Tables, equations, and constants

  1. The pairs (2, 7), (5, 17.5), and (8, 28) are given. Is the relationship proportional? If yes, find k.
  2. The pairs (3, 4), (6, 8), and (9, 13) are given. Is the relationship proportional? Explain.
  3. A proportional relationship contains the point (4, 18). Find the constant of proportionality.
  4. Seven notebooks cost $10.50. Write an equation for cost c in terms of notebooks n.
  5. Use y = 2.4x to find y when x = 7.5.
  6. In y = 5x, explain the meaning of the point (6, 30).

C. Graphs and context

  1. A proportional graph passes through (0, 0) and (4, 14). Find k and write the equation.
  2. A proportional graph contains the point (1, 2.75). What does 2.75 represent?
  3. Six event tickets cost $48. Write a proportional equation for total cost c and ticket count t.
  4. Plan A costs $3.75 per gigabyte. Plan B costs $22.50 for 5 gigabytes. Which plan has the lower unit rate?

D. Percent applications

  1. A $90 item is discounted by 20%. Find the sale price.
  2. A meal costs $52 before an 18% tip. Find the total including tip.
  3. A store marks up an $80 item by 35%. Find the selling price.
  4. A salesperson earns a $300 base payment plus 7% commission on $6,400 in sales. Find total pay.
  5. Find the simple interest on $750 at 4.5% per year for 2 years.
  6. A measurement is 102 cm while the actual length is 100 cm. Find the percent error.

Complete all 20 questions before opening the answers.

Go to the Answer Key
Check your work

Answer key

Use each explanation to identify the skill or representation that needs more practice.

Question 112 miles per hour

9 ÷ 3/4 = 9 × 4/3 = 12.

Question 23 1/3 cups per batch

(5/2) ÷ (3/4) = 10/3.

Question 3$9 per pound

7.50 ÷ (5/6) = 7.50 × 6/5 = 9.

Question 41/2 gallon per wall

(3/8) ÷ (3/4) = 1/2.

Question 5Yes; k = 3.5

Each y ÷ x ratio equals 3.5.

Question 6No

4/3 = 8/6, but 13/9 is different.

Question 7k = 4.5

18 ÷ 4 = 4.5.

Question 8c = 1.5n

10.50 ÷ 7 = 1.50 per notebook.

Question 918

2.4 × 7.5 = 18.

Question 10Six units of x correspond to 30 units of y

The point satisfies y = 5x.

Question 11k = 3.5; y = 3.5x

14 ÷ 4 = 3.5.

Question 12The unit rate or constant of proportionality

At x = 1, y equals 2.75.

Question 13c = 8t

48 ÷ 6 = 8 dollars per ticket.

Question 14Plan A

$22.50 ÷ 5 = $4.50 per GB, which is more than $3.75.

Question 15$72

90 × 0.80 = 72.

Question 16$61.36

Tip = 52 × 0.18 = 9.36; total = 61.36.

Question 17$108

80 × 1.35 = 108.

Question 18$748

Commission = 0.07 × 6,400 = 448; add $300.

Question 19$67.50

I = 750 × 0.045 × 2 = 67.50.

Question 202%

|102 − 100| ÷ 100 × 100% = 2%.

Focused review

A five-session proportional relationships plan

Session 1

Fractional unit rates

Divide fractional quantities and state units clearly.

Session 2

Tables and constants

Test y ÷ x and identify k from tables and contexts.

Session 3

Graphs and equations

Connect origin, slope, (1, r), and y = kx.

Session 4

Percent applications

Practice increase, decrease, tax, tips, commission, interest, and error.

Session 5

Mixed transfer

Complete the 20-question set, correct errors, and retry weak types.

Continue the Grade 7 sequence

Use the Grade 7 Math Skills Checklist, the Free Grade 7 Diagnostic, and the Grade 7 Complete Revision Pack to connect this topic to the full curriculum.

Continue to Rational Numbers Review
Frequently asked questions

Grade 7 proportional relationships FAQ

What is a proportional relationship in Grade 7 math?

A proportional relationship is a relationship between two quantities with a constant ratio. It can be represented by an equation of the form y = kx, where k is the constant of proportionality.

How can a student test whether a table is proportional?

Divide each y-value by its matching x-value. The relationship is proportional when the ratio y divided by x remains constant for every nonzero x-value.

What does the point (1, r) mean on a proportional graph?

The point (1, r) shows the unit rate. It means that when x equals 1, y equals r.

Why must a proportional graph pass through the origin?

In y = kx, substituting x = 0 gives y = 0. Therefore, every proportional relationship includes the point (0, 0).

What percent problems are included in Grade 7?

Grade 7 percent work commonly includes tax, tips, discounts, markups, markdowns, commissions, fees, percent increase and decrease, simple interest, and percent error.

How should a student study proportional relationships?

Practice moving among tables, graphs, equations, and contexts. Write the unit rate or constant first, then explain what it means before completing the calculation.

Standards reference and scope

This guide is organized around the Grade 7 Ratios and Proportional Relationships standards for fractional unit rates, proportional tables and graphs, equations, point interpretation, and multi-step percent problems. Read the official Common Core mathematics standards. GradeCove is an independent educational publisher and is not endorsed by the Common Core State Standards Initiative.

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